Sometimes the normals on a mesh can import looking tremendously glitched out. It can manifest as faces appearing black, as if the normals are facing the wrong way, or odd chrome-like reflections, or other shading artifacts.

Maya has a suite of tools for visualizing and editing normals, but if everything checks out cleanly in Maya then something else can be the culprit: The Construction History.

For a static mesh (one without a skinning deformer), select the mesh and use Edit->Delete By Type->History and freeze the construction history.

For a skinned mesh, like a character, use Delete by Type->Non-Deformer History instead, otherwise it will delete the connection to the joints and it won't animate any more.